Nil 3-Armendariz Rings

Show more

We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring *R* is nil 3-Armendariz ring if and only if for any , *T*_{n}(*R*) is nil 3-Armendariz ring. Also we prove that a right Ore ring *R* is nil 3-Armendariz if and only if so is *Q*, where *Q* is the classical right quotient ring of *R*. With the help of this result, we can show that a commutative ring *R* is nil 3-Armendariz if and only if the total quotient ring of *R* is nil 3-Armendariz.

References

[1] E. P. Armendariz, “A Note on Extensions of Baer and p.p.-Rings,” Journal of the Australian Mathematical Society, Vol. 18, No. 4, 1974, pp. 470-473.

http://dx.doi.org/10.1017/S1446788700029190

[2] M. B. Rege and S. Chhawchharia, “Armendariz Rings,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 1, 1997, pp. 14-17.

http://dx.doi.org/10.3792/pjaa.73.14

[3] D. D. Anderson and V. Camillo, “Armendariz Rings and Gaussian Rings,” Communications in Algebra, Vol. 26, No. 7, 1998, pp. 2265-2272.

http://dx.doi.org/10.1080/00927879808826274

[4] N. K. Kim and Y. Lee, “Armendariz Rings and Reduced Rings,” Journal of Algebra, Vol. 223, No. 2, 2000, pp. 477-488. http://dx.doi.org/10.1006/jabr.1999.8017

[5] C. Huh, Y. Lee and A. Smoktunowicz, “Armendariz Rings and Semicommutative Rings,” Communications in Algebra, Vol. 30, No. 2, 2002, pp. 751-761.

http://dx.doi.org/10.1006/jabr.1999.8017

[6] T. K. Lee and T. L. Wong, “On Armendariz Rings,” Houston Journal of Mathematics, Vol. 29, No. 3, 2003, pp. 583-593.

[7] Y. Suiyi, “On the Extension of Armendariz Rings,” MS Thesis, Lanzhou University, Lanzhou, 2008.

[8] R. Antoine, “Nilpotent Elements and Armendariz Rings,” Journal of Algebra, Vol. 319, No. 8, 2008, pp. 3128-3140.

http://dx.doi.org/10.1016/j.jalgebra.2008.01.019

[9] Z. K. Liu and R. Y. Zhao, “On Weak Armendariz Rings,” Communications in Algebra, Vol. 34, No. 7, 2006, pp. 2607-2616. http://dx.doi.org/10.1080/00927870600651398

[10] H. F. Wu, “On Weak 3-Armendariz Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 11, No. 3, 2012, pp. 241-244.

[11] H. F. Wu, “Extensions of Reduced Rings,” Journal of Hangzhou Normal University (Natural Science Edition), Vol. 10, No. 5, 2011, pp. 407-410.

[12] J. C. McConnell and J. C. Robson, “Noncommutative Noetherian Rings,” A Wiley-Interscience Publication, John Wiley and Sons, Ltd., Chichester, 1987.